While I disagree that “billions is beyond being halved”, there is some truth to the idea that numbers can get so big that halving doesn’t make much difference. That seems very very counter-intuitive, so I’ll try to briefly explain.
Consider (10^10 + 2). That’s 10000000000+2. I think it’s fair to say that the +2 doesn’t make a lot of difference. It’s still approximately 10^10.
So then, consider 10^(10^10)×100. That’s a huge number, too big to type here, then multiplied by 100. So the result is 100 times bigger than the huge number. But… writing it down we see this:
10^(10^10)×100 = 10^(10^10+2) ≈ 10^(10^10).
So although ×100 does make it one hundred times bigger… that just doesn’t really make a lot of difference to a number as big as that one. As numbers get bigger and bigger, they start to take on properties a bit like ‘infinity’. Addition stops being important, then multiplication, then for even bigger numbers exponentiation doesn’t huge much of an impact either.
Mathematically, I think this is really cool and interesting. But I don’t think 1 billion is that big. 10^9 is big enough that +2 doesn’t matter much, but not so big that ×2 doesn’t matter.
[edit] (I’m struggling to get the nested powers to look right… So hopefully my meaning is clear enough anyway.)
While I disagree that “billions is beyond being halved”, there is some truth to the idea that numbers can get so big that halving doesn’t make much difference. That seems very very counter-intuitive, so I’ll try to briefly explain.
Consider (10^10 + 2). That’s 10000000000+2. I think it’s fair to say that the +2 doesn’t make a lot of difference. It’s still approximately 10^10.
So then, consider 10^(10^10)×100. That’s a huge number, too big to type here, then multiplied by 100. So the result is 100 times bigger than the huge number. But… writing it down we see this:
10^(10^10)×100 = 10^(10^10+2) ≈ 10^(10^10).
So although ×100 does make it one hundred times bigger… that just doesn’t really make a lot of difference to a number as big as that one. As numbers get bigger and bigger, they start to take on properties a bit like ‘infinity’. Addition stops being important, then multiplication, then for even bigger numbers exponentiation doesn’t huge much of an impact either.
Mathematically, I think this is really cool and interesting. But I don’t think 1 billion is that big. 10^9 is big enough that +2 doesn’t matter much, but not so big that ×2 doesn’t matter.
[edit] (I’m struggling to get the nested powers to look right… So hopefully my meaning is clear enough anyway.)
Maybe posting a picture where you write it out? I think I understand though.
Interesting concept, thank you!
Have you ever played idle champions of the forgotten realms? At some point you start to get this feeling.
Happens in MMOs too, world of Warcraft was bad with it over the years on and off.
They’ve reduced the numbers down multiple times so that people start to feel progression again.